This phase II trial studies how well durvalumab works in treating patients with stage IV squamous cell lung cancer that has come back after previous treatment. This is a "non-match" sub-study that includes all screened patients not eligible for a biomarker-driven sub-study. Monoclonal antibodies, such as durvalumab, may be able to shrink tumors. Durvalumab may be effective in treating patients with squamous cell lung cancer.